HAL Id: hal-01919518
https://hal.sorbonne-universite.fr/hal-01919518
Preprint submitted on 12 Nov 2018HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Modulo p representations of reductive p-adic groups:
functorial properties
Noriyuki Abe, Guy Henniart, Marie-France Vignéras
To cite this version:
Noriyuki Abe, Guy Henniart, Marie-France Vignéras. Modulo p representations of reductive p-adic groups: functorial properties. 2018. �hal-01919518�
MODULO p REPRESENTATIONS OF REDUCTIVE p-ADIC GROUPS: FUNCTORIAL PROPERTIES
N. ABE, G. HENNIART, AND M.-F. VIGN ´ERAS
Abstract. Let F be a local field with residue characteristic p, let C be an algebraically closed field of characteristic p, and let G be a connected reductive F -group. In a previous paper, Florian Herzig and the authors classified irreducible admissible C-representations of G = G(F ) in terms of supercuspidal representations of Levi subgroups of G. Here, for a parabolic subgroup P of G with Levi subgroup M and an irreducible admissible C-representation τ of M , we determine the lattice of subC-representations of IndG
Pτ and we show
that IndGPχτ is irreducible for a general unramified character χ of M . In the reverse direction,
we compute the image by the two adjoints of IndG
P of an irreducible admissible representation
π of G. On the way, we prove that the right adjoint of IndGP respects admissibility, hence
coincides with Emerton’s ordinary part functor OrdG
P on admissible representations.
Contents
1. Introduction 2
1.1. Classification results of [AHHV17] 2
1.2. Main results 2
1.3. Outline of the proof 3
Acknowledgment 4
2. Notation, useful facts and preliminaries 4
2.1. The group G and its standard parabolic subgroups P = M N 4
2.2. Representations of G 5
2.3. On compact induction 5
2.4. IG(P, σ, Q) and minimality 6
2.5. Hecke algebras 8
3. Lattice of subrepresentations of IndGPσ, σ irreducible admissible 8
3.1. Result 8
3.2. Proof 9
3.3. Twists by unramified characters 12
4. Admissibility 12
4.1. Generalities 12
4.2. Examples 13
4.3. Admissibility and RGP 15
4.4. Admissibility of IG(P, σ, Q) 19
4.5. IndGP does not respect finitely generated representations 19 5. Composing IndGP with adjoints of IndGP
1 when p is nilpotent 20
5.1. Results 20
2010 Mathematics Subject Classification. primary 20C08, secondary 11F70.
The first-named author was supported by JSPS KAKENHI Grant Number 26707001.
5.2. Proofs 24 6. Applying adjoints of IndGP
1 to IG(P, σ, Q) 25
6.1. Results and applications 25
6.2. The case of N1-coinvariants 28
7. Ordinary functor OrdG
P1 29
7.1. Haar measure and t-finite elements 29
7.2. Filtrations 32 7.3. Case P1 ⊃ P 35 7.4. Case hP, P1i = G 37 7.5. General case 38 References 39 1. Introduction
1.1. Classification results of [AHHV17]. The present paper is a sequel to [AHHV17]. The overall setting is the same: p is a prime number, F a local field with finite residue field of characteristic p, G a connected reductive F -group and G = G(F ) is seen as a topological locally pro-p group. We fix an algebraically closed field C of characteristic p and we study the smooth representations of G over C-vector spaces - we write Mod∞C(G) for the category they form.
Let P be a parabolic subgroup of G with a Levi decomposition P = M N and σ a su-percuspidal C-representation of M , in the sense that it is irreducible, admissible, and does not appear as a subquotient of a representation of M obtained by parabolic induction from an irreducible, admissible C-representation of a proper Levi sugroup of M . Then there is a maximal parabolic subgroup P (σ) of G containing P to which σ inflated to P extends; we write e(σ) for that extension. For each parabolic subgroup Q of G with P ⊂ Q ⊂ P (σ), we form
IG(P, σ, Q) = IndGP (σ)(e(σ) ⊗ St P (σ) Q ) where StP (σ)Q = IndP (σ)Q 1/P
IndP (σ)Q0 1, the sum being over parabolic subgroups Q0 of G with
Q ( Q0 ⊂ P (σ).
The classification result of [AHHV17] is that IG(P, σ, Q) is irreducible admissible, and that conversely any irreducible admissible C-representation of G has the form IG(P, σ, Q), where
P is determined up to conjugation, and, once P is fixed, Q is determined and so is the
isomorphism class of σ.
1.2. Main results. The classification raises natural questions: if G is a Levi subgroup of a parabolic subgroup R in a larger connected reductive group H, what is the structure of IndHRπ when π is a irreducible admissible C-representation of G?
We show that IndHRπ has finite length and multiplicity 1; we determine its irreducible
constituents and the lattice of its subrepresentations: see section 3 for precise results and proofs. As an application, we answer a question of Jean-Francois Dat, in showing that IndHRχπ
is irreducible when χ is a general unramified character of G.
If P1 is a parabolic subgroup of G with Levi decomposition P1 = M1N1, then IndGP1 :
Mod∞C(M1) → Mod∞C(G) has a left adjoint LGP1, which is the usual Jacquet functor (−)N1
LG
P1 and R
G
P1 to π. They turn out to be irreducible or 0, in sharp contrast to the case of
complex representations of G. To state precise results, we fix a minimal parabolic subgroup
B of G and a Levi decomposition B = ZU of B, and we consider only parabolic subgroups
containing B and their Levi components containing Z. We simply say “let P = M N be a standard parabolic subgroup of G” to mean that P contains B and M is the Levi component of P containing Z, N being the unipotent radical of P .
Theorem 1.1. Let P = M N and P1 = M1N1 be standard parabolic subgroups of G, let σ be a supercuspidal C-representation of M and let Q be a parabolic subgroup of G with P ⊂ Q ⊂ P (σ).
(i) LG
P1IG(P, σ, Q) is isomorphic to IM1(P ∩ M1, σ, Q ∩ M1) if P1 ⊃ P and the group
generated by P1∪ Q contains P (σ), and is 0 otherwise.
(ii) RPG
1IG(P, σ, Q) is isomorphic to IM1(P ∩ M1, σ, Q ∩ M1) if P1 ⊃ Q, and is 0 otherwise.
See §6 and §7 for the proofs, with consequences already drawn in §6.1: in particular, we prove that an irreducible admissible C-representation π of G is supercuspidal exactly when
LGPπ and RGPπ are 0 for any proper parabolic subgroup P of G.
As the construction of IG(P, σ, Q) involves parabolic induction, we are naturally led to investigate, as an intermediate step, the composite functors LGP
1Ind
G
P and RGP1Ind
G
P, for stan-dard parabolic subgroups P = M N and P1= M1N1 of G. In §5, we prove:
Theorem 1.2. The functor LGP
1Ind
G
P : Mod∞C(M ) → Mod∞C(M1) is isomorphic to the functor
IndM1
P ∩M1L
M
P1∩M, and the functor R
G P1Ind
G
P : Mod∞C(M ) → Mod∞C(M1) is isomorphic to the functor IndM1
P ∩M1R
M P1∩M.
We actually describe explicitly the functorial isomorphism for LGP
1Ind
G
P whereas the case of RGP
1Ind
G
P is obtained by adjunction properties. The fact that RGP1 has no direct explicit
description has consequence for the proof of Theorem 1.1 (ii). We first prove:
Theorem 1.3. If π is an admissible C-representation of G, then RPGπ is an admissible C-representation of M .
It follows that on admissible C-representations of G, RGP coincides with Emerton’s ordi-nary part functor OrdG
P (as extended to the case of C-representations in [Vig13]). To prove Theorem 1.1 (ii) we in fact use OrdG
P1 in place of R
G
P1. Note that, if the characteristic of F
is 0 and π is an admissible C-representation of G, then LGPπ is admissible. But in contrast,
when F has characteristic p, we produce in §4 an example, for G = SL(2, F ), of an admissible
C-representation π of G such that LGBπ is not admissible.
1.3. Outline of the proof. After the initial section §2 devoted to notation and preliminaries, our paper mainly follows the layout above. However admissibility questions are explored in §4, where Theorem 1.3 is established: as mentioned above, the result is used in the proof Theorem 1.1 (ii).
Without striving for the utmost generality, we have taken care not to use unnecessary assumptions. In particular, from section §4 on, we consider a general commutative ring R as coefficient ring, imposing conditions on R only when useful. The reason is that for arithmetic applications it is important to consider the case where R is artinian and p is nilpotent or invertible in R. Only when we use the classification do we assume R = C. Our results are valid for R noetherian and p nilpotent in R in sections §4 to §7. For example, when R is
noetherian and p is nilpotent in R, Theorem 1.2 is valid (Theorem 5.5 and Corollary 5.6) and a version to Theorem 1.1 is obtained in Theorem 6.1 and Corollary 6.2. Likewise Theorem 1.3 is valid when R is noetherian and p is nilpotent in R (Theorem 4.11).
In a companion paper [AHV], the authors will investigate the effect of taking invariants under a pro-p Iwahori subgroup in the modules IG(P, σ, Q) of 1.1.
Acknowledgment. The authors thank the referee for a thorough reading and helpful com-ments. They also thank Julien Hauseux pointing out an error in Proposition 4.22.
2. Notation, useful facts and preliminaries
2.1. The group G and its standard parabolic subgroups P = M N . In all that follows,
p is a prime number, F is a local field with finite residue field k of characteristic p; as usual,
we write OF for the ring of integers of F , PF for its maximal ideal and valF the absolute value of F normalised by valF(F∗) = Z. We denote an algebraic group over F by a bold letter, like H, and use the same ordinary letter for the group of F -points, H = H(F ). We fix a connected reductive F -group G. We fix a maximal F -split subtorus T and write Z for its G-centralizer; we also fix a minimal parabolic subgroup B of G with Levi component Z, so that B = ZU where U is the unipotent radical of B. Let X∗(T) be the group of F -rational characters of T and Φ the subset of roots of T in the Lie algebra of G. Then B determines a subset Φ+ of positive roots - the roots of T in the Lie algebra of U- and a subset of simple roots ∆. The G-normalizer NG of T acts on X∗(T) and through that action, NG/Z identifies with the
Weyl group of the root system Φ. Set N := NG(F ) and note that NG/Z ' N /Z; we write
W for N /Z.
A standard parabolic subgroup of G is a parabolic F -subgroup containing B. Such a parabolic subgroup P has a unique Levi subgroup M containing Z, so that P = MN where N is the unipotent radical of P - we also call M standard. By a common abuse of language to describe the preceding situation, we simply say “let P = M N be a standard parabolic subgroup of G”; we sometimes write NP for N and MP for M . The parabolic subgroup of G opposite to P will be written P and its unipotent radical N , so that P = M N , but beware that P is not standard ! We write WM for the Weyl group M ∩ N /Z.
If P = MN is a standard parabolic subgroup of G, then M ∩ B is a minimal parabolic subgroup of M. If ΦM denotes the set of roots of T in the Lie algebra of M, with respect to M ∩ B we have Φ+M = ΦM ∩ Φ+ and ∆M = ΦM ∩ ∆. We also write ∆P for ∆M as P and M determine each other, P = M U . Thus we obtain a bijection P 7→ ∆P from standard parabolic subgroups of G to subsets of ∆, with B corresponds to ∅ and G to ∆. If I is a subset of ∆, we sometimes denote by PI= MINI the corresponding standard parabolic subgroup of G. If I = {α} is a singleton, we write Pα = MαNα. We note a few useful properties. If P1
is another standard parabolic subgroup of G, then P ⊂ P1 if and only if ∆P ⊂ ∆P1; we have
∆P ∩P1 = ∆P ∩ ∆P1 and the parabolic subgroup corresponding to ∆P ∪ ∆P1 is the subgroup hP, P1i of G generated by P and P1. The standard parabolic subgroup of M associated to
∆M∩∆M1 is M ∩ P1= (M ∩ M1)(M ∩ N1) [Car85, Proposition 2.8.9]. It is convenient to write
G0 for the subgroup of G generated by the unipotent radicals of the parabolic subgroups; it is also the normal subgroup of G generated by U , and we have G = ZG0.
For each α ∈ X∗(T), the homomorphism x 7→ valF(α(x)) : T → Z extends uniquely to a homomorphism Z → Q that we denote in the same way. This defines a homomorphism
An interesting situation occurs when ∆ = I tJ is the union of two orthogonal subsets I and
J . In that case, G0 = MI0MJ0, MI0 and MJ0 commute with each other, and their intersection is finite and central in G [AHHV17, II.7 Remark 5].
2.2. Representations of G. As apparent in the abstract and the introduction, our main interest lies in smooth C-representations of G, where C is an algebraically closed field of characteristic p, which we fix throughout. However many of our arguments do not necessitate so strong a hypothesis on coefficients, so we let R be a fixed commutative ring. Occasionally we shall consider an R[A]-module V where A is a monoid. An element v of V is called A-finite if its translates under A generate a finitely generated submodule of V . If
R is noetherian the A-finite elements in V generate a submodule of V , that we write VA−f. When A is generated by an element t, we write Vt−f instead of VA−f.
We speak indifferently of R[H]-modules and of R-representations of H for a locally profinite group H. An R[H]-module V is called smooth if every vector in V has an open stabilizer in H. The smooth R-representations of H and R[H]-linear maps form an abelian category Mod∞R(H).
An R-representation V of a locally profinite group H is admissible if it is smooth and for any open compact subgroup J of H, the R-submodule VJ of J -fixed vectors is finitely generated. When R is noetherian, it is clear that it suffices to check this when J is small enough. When R is noetherian we write ModaR(H) for the subcategory of Mod∞R(H) made out of the admissible R-representations of H. We explore admissibility further in section 4.
If P = M N is a standard parabolic subgroup of G, the parabolic induction functor IndGP : Mod∞R(M ) → Mod∞R(G) sends W ∈ Mod∞R(M ) to the smooth R[G]-module IndGP W made
out of functions f : G → W satisfying f (mngk) = mf (g) for m ∈ M, n ∈ N, g ∈ G and k in some open subgroup Kf of G - the action of G is via right translation. The functor IndGP has a left adjoint LGP : Mod∞R(G) → Mod∞R(M ) which sends V in Mod∞R(G) to the module of
N -coinvariants VN of V , which is naturally a smooth R[M ]-module. The functor IndGP has a right adjoint RGP : Mod∞R(G) → Mod∞R(M ) [Vig13, Proposition 4.2].
When R is a field, a smooth R-representation of G is called irreducible if it is a simple
R[G]-module. An R-representation of G is called supercuspidal it is irreducible, admissible,
and does not appear as a subquotient of a representation of M obtained by parabolic induction from an irreducible, admissible representation of a proper Levi subgroup of M .
2.3. On compact induction. If X is a locally profinite space with a countable basis of open sets, and V is an R-module, we write Cc∞(X, V ) for the space of compactly supported locally constant functions X → V . One verifies that the natural map Cc∞(X, R) ⊗RV → Cc∞(X, V ) is an isomorphism.
Lemma 2.1. The R-module Cc∞(X, R) is free. When X is compact, the submodule of constant
functions is a direct factor of Cc∞(X, R).
Proof. The proof of [Ly15, Appendix A.1] when X is compact is easily adapted to Cc∞(X, V )
when X is not compact.
Example 2.2. Cc∞(X, R)H is a direct factor of Cc∞(X, R) when X is compact with a continuous action of a profinite group H with finitely many orbits (apply the lemma to the orbits which are open).
Lemma 2.3. The quotient map H → J \H has a continuous section.
Proof. When H is profinite, this is [RZ10, Proposition 2.2.2]. In general, let K be a compact
open subgroup of H. Cover H with disjoint double cosets J gK. It is enough to find, for any given g, a continuous section of the induced map J gK−→ J \J gK. The map k 7→ gk inducesπg a continous bijective map (K ∩ g−1J g)\K −→ J \J gK. Because J is closed in H, both spacesp are Hausdorff and (K ∩ g−1J g)\K is compact since K is, so p is a homeomorphism. If σ is a
continuous section of the quotient map K → (K ∩ g−1J g)\K then x 7→ gσ(p−1(x)) gives the
desired section of πg.
Let σ be a continuous section of H → J \H, and let V be a smooth R-representation of J . Recall that c-IndHJ V is the space of functions f : H → V , left equivariant by J , of compact
support in J \H, and smooth for H acting by right translation. Immediately:
Lemma 2.4. The map f 7→ f ◦ σ : c-IndHJ V → Cc∞(J \H, V ) is an R-module isomorphism. As a consequence we get a useful induction/restriction property: let W be a smooth R-representation of H.
Lemma 2.5. The map f ⊗ w 7→ (h 7→ f (h) ⊗ hw) : (c-IndHJ V ) ⊗ W → c-IndHJ (V ⊗ W ) is an
R[H]-isomorphism.
Proof. The map is linear and H-equivariant. Lemma 2.4 implies that it is bijective. Remark 2.6. Arens’ theorem says that if X is a homogeneous space for H and H/K is
countable for a compact open subgroup K of H, then for x ∈ X the orbit map h 7→ hx induces a homeomorphism H/Hx' X. In particular, for two closed subgroups I, J of H such that H = IJ , we get a homeomorphism I/(I ∩ J ) ' H/J . Hence (c-IndHJ V )|I ' c-IndII∩JV for any smooth R-representation V of J .
2.4. IG(P, σ, Q) and minimality. We recall from [AHHV17] the construction of IG(P, σ, Q), our main object of study.
Proposition 2.7. Let P = M N ⊂ Q be two standard parabolic subgroups of G and σ an
R-representation of M . Then the following are equivalent:
(i) σ extends to a representation of Q where N acts trivially. (ii) For each α ∈ ∆Q\ ∆P, Z ∩ Mα0 acts trivially on σ.
That comes from [AHHV17, II.7 Proposition] when R = C, but the result is valid for any commutative ring R [AHHV17, II.7 first remark 2]. Besides, the extension of σ to Q, when the conditions are fulfilled, is unique; we write it eQ(σ); it is trivial on NQ and we view it equally as a representation of MQ. The R-representation eQ(σ) of Q or MQ is smooth, or admissible, or irreducible (when R is a field) if and only if σ is. Let Pσ = MσNσ be the standard parabolic subgroup of G with ∆Pσ = ∆σ where
(1) ∆σ = {α ∈ ∆ \ ∆P | Z ∩ Mα0 acts trivially on σ}.
There is a largest parabolic subgroup P (σ) containing P to which σ extends: ∆P (σ) = ∆P ∪ ∆σ. Clearly when P ⊂ Q ⊂ P (σ), the restriction to Q of eP (σ)(σ) is eQ(σ). If there is no risk of ambiguity, we write
Definition 2.8. An R[G]-triple is a triple (P, σ, Q) made out of a standard parabolic sub-group P = M N of G, a smooth R-representation of M , and a parabolic subsub-group Q of G with P ⊂ Q ⊂ P (σ). To an R[G]-triple (P, σ, Q) is associated a smooth R-representation of
G:
IG(P, σ, Q) = IndGP (σ)(e(σ) ⊗ St P (σ) Q )
where StP (σ)Q is the quotient of IndP (σ)Q 1, 1 denoting the trivial R-representation of Q, by the sum of its subrepresentations IndP (σ)Q0 1, the sum being over the set of parabolic subgroups Q0
of G with Q ( Q0 ⊂ P (σ).
Note that IG(P, σ, Q) is naturally isomorphic to the quotient of IndGQ(eQ(σ)) by the sum of its subrepresentations IndGQ0(eQ0(σ)) for Q ( Q0 ⊂ P (σ) by Lemma 2.5.
We also remark that we have the identifications IndQP σ ' IndQ/NQ
P /NQσ and St
Q P ' St
Q/NQ
P /NQ
where P ⊂ Q are parabolic subgroups, NQthe unipotent radical of Q and σ an representation of P with the trivial action of NP (hence a representation of the Levi quotient of P ). The subgroup P/NQ of Q/NQ is a parabolic subgroup.
It might happen that σ itself has the form eP(σ1) for some standard parabolic subgroup
P1 = M1N1 contained in P and some R-representation σ1 of M1. In that case, P (σ1) = P (σ)
and e(σ) = e(σ1). We say that σ is e-minimal if σ = eP(σ1) implies P1 = P, σ1 = σ.
Lemma 2.9. Let P = M N be a standard parabolic subgroup of G and let σ be an
R-representation of M . There exists a unique standard parabolic subgroup Pmin,σ = Mmin,σNmin,σ of G and a unique e-minimal representation of σmin of Mmin,σ with σ = eP(σmin). Moreover P (σ) = P (σmin) and e(σ) = e(σmin).
Proof. We have
(2) ∆Pmin,σ = {α ∈ ∆P | Z ∩ M
0
α does not act trivially on σ}, σmin is the restriction of σ to Mmin,σ, and
(3) ∆σmin = {α ∈ ∆ | Z ∩ Mα0 acts trivially on σ}.
Lemma 2.10. Let P = M N be a standard parabolic subgroup of G and σ an e-minimal
R-representation of M . Then ∆P and ∆σ are orthogonal.
That comes from [AHHV17, II.7 Corollary 2]. That corollary of loc. cit. also shows that when R is a field and σ is supercuspidal, then σ is e-minimal. Lemma 2.10 shows that ∆Pmin,σ and ∆σmin are orthogonal.
Note that when ∆P and ∆σ are orthogonal of union ∆ = ∆Pt∆σ, then G = P (σ) = M Mσ0 and e(σ) is the R-representation of G simply obtained by extending σ trivially on Mσ0. Lemma 2.11. Let (P, σ, Q) be an R[G]-triple. Then (Pmin,σ, σmin, Q) is an R[G]-triple and IG(P, σ, Q) = IG(Pmin,σ, σmin, Q).
2.5. Hecke algebras. We fix a special parahoric subgroup K of G fixing a special vertex
x0 in the apartment A associated to T in the Bruhat-Tits building of the adjoint group of G. If V is an irreducible smooth C-representation of K, we have the compactly induced
representation c-IndGKV of G, its endomorphism algebra HG(K, V ) and the centre ZG(K, V ) of HG(K, V ). For a standard parabolic subgroup P = M N of G, the group M ∩ K is a special parahoric subgroup of M and VN ∩K is an irreducible smooth C-representation of M ∩ K. For W ∈ Mod∞C(M ), there is an injective algebra homomorphism
SG
P : HG(K, V ) → HM(M ∩ K, VN ∩K)
for which the natural isomorphism HomG(c-IndGKV, IndGP W ) ' HomM(c-IndMM ∩KVN ∩K, W ) is SPG-equivariant [HV15], [HV12]. Moreover. SPG(ZG(K, V )) ⊂ ZM(M ∩ K, VN ∩K).
Let Z(M ) denote the maximal split central subtorus of M ; it is equal to the group of
F -points of the connected component in T of T
α∈∆MKer α. Let z ∈ Z(M ). We say that
z strictly contracts an open compact subgroup N0 of N if the sequence (zkN0z−k)k∈N is strictly decreasing of intersection {1}. We say that z strictly contracts N if there exists an open compact subgroup N0 ⊂ N such that z strictly contracts N0. Choose z ∈ Z(M ) which
strictly contracts N . Let τ ∈ ZM(M ∩ K, VN ∩K) be a non-zero element which supports on (M ∩K)z(M ∩K). (Such an element is unique up to constant multiplication.) Then τ ∈ Im SPG and the algebra HM(K ∩ M, VN ∩K) (resp. ZM(M ∩ K, VN ∩K)) is the localization of HG(K, V ) (resp. ZG(K, V )) at τ .
3. Lattice of subrepresentations of IndGPσ, σ irreducible admissible 3.1. Result. This section is a direct complement to [AHHV17]. Our coefficient ring is R = C. We are given a standard parabolic subgroup P1 = M1N1of G and an irreducible admissible
C-representation σ1 of M1. Our goal is to describe the lattice of subrepresentations of IndGP
1σ1.
We shall see that IndGP
1σ1has finite length and is multiplicity free, meaning that its irreducible
constituents occur with multiplicity 1. We recall the main result of [AHHV17] :
Theorem 3.1 (Classification Theorem). (A) Let P = M N be a standard parabolic subgroup
of G and σ a supercuspidal C-representation of M . Then IndGPσ ∈ Mod∞C(G) has finite
length and is multiplicity free of irreducible constituents the representations IG(P, σ, Q) for P ⊂ Q ⊂ P (σ), and all IG(P, σ, Q) are admissible.
(B) Let π be an irreducible admissible C-representation of G. Then, there is a C[G]- triple
(P, σ, Q) with σ supercuspidal, such that π is isomorphic to IG(P, σ, Q) and π determines P, Q and the isomorphism class of σ.
By the classification theorem, there is a standard parabolic subgroup P = M N of G and a supercuspidal C-representation σ of M such that σ1 occurs in IndMP ∩M1 1σ. More precisely, if P (σ) is the largest standard parabolic subgroup of G to which σ extends, then by Proposition
2.7, P (σ) ∩ M1 is the largest standard parabolic subgroup of M1 to which σ extends and σ1 ' IM1(P ∩ M1, σ, Q) ' Ind
M1
P (σ)∩M1(eP (σ)∩M1(σ) ⊗ St
P (σ)∩M1
Q )
for some parabolic subgroup Q of M1 with (P ∩ M1) ⊂ Q ⊂ (P (σ) ∩ M1). By transitivity of
the parabolic induction, IndGP 1σ1' Ind G P (σ)(e(σ) ⊗ Ind M (σ) M (σ)∩P1St P (σ)∩M1 Q ),
and we need to analyse this representation. Our analysis is based on [Her11, §10]. We recall the structure of the lattice of subrepresentations of a finite length multiplicity free representation X. Let J be the set of its irreducible constituents. For j ∈ J , there is a unique subrepresentation Xj of X with cosocle j - it is the smallest subrepresentation of X with j as a quotient. Put the order relation ≤ on J , where i ≤ j if i is a constituent of Xj. Then the lattice of subrepresentations of X is isomorphic to the lattice of lower sets in (J, ≤) - recall that such a lower set is a subset J0 of J such that if j1∈ J, j2 ∈ J0 and j1 ≤ j2 then j1 ∈ J0. A subrepresentation of X is sent to the lower set made out of its irreducible constituents, and a lower set J0 of J is sent to the sum of the subrepresentations Xj for j ∈ J0. We have Xj = j if and only if j is minimal in (J, ≤). If the cosocle of X is irreducible, then (J, ≤) has the unique maximal element and Xj = X if and only if j is maximal in (J, ≤). The socle of X is the direct sum of the minimal j ∈ (J, ≤) and the cosocle of X is the direct sum of the
maximal j ∈ (J, ≤).
In the sequel J will often be identified with P(I) for some subset I of ∆, both equipped with the order relation reverse to the inclusion. Thus we rather talk of upper sets in P(I) (for the inclusion). In that case the socle I of X and the cosocle ∅ of X are both irreducible. Theorem 3.2. With the above notations, IndGP
1σ1 has finite length and is multiplicity free,
of irreducible constituents the IG(P, σ, Q0) where Q0 is a parabolic subgroup of G satisfying P ⊂ Q0 ⊂ P (σ) and ∆P1 ∩ ∆Q0 = ∆Q. Sending IG(P, σ, Q
0) to ∆
Q0 ∩ (∆ \ ∆P
1) gives an
isomorphism of the lattice of subrepresentations of IndGP
1σ1 onto the lattice of upper sets in
P(∆P (σ)∩ (∆ \ ∆P1)).
The first assertion is a consequence of the classification theorem 3.1 since IndGP1σ1 is a
subrepresentation of IndGPσ. For the rest of the proof, given in §3.2, we proceed along the
classification, treating cases of increasing generality. As an immediate consequence of the theorem, we get an irreducibility criterion.
Corollary 3.3. The representation IndGP
1σ1 is irreducible if and only if P1 contains P (σ).
Corollary 3.4. The socle and the cosocle of IndGP
1σ1 are both irreducible.
This is very different from the complex case [LM16].
3.2. Proof. We proceed now to the proof of Theorem 3.2. The very first and basic case is when P1 = B and σ1 is the trivial representation 1 of Z. The irreducible constituents of
IndGB1 are the StGQ for the different standard parabolic subgroups Q of G, each occuring with multiplicity 1.
Proposition 3.5. Let Q be a standard parabolic subgroup of G. (i) The submodule of IndGB1 with cosocle StGQ is IndGQ1.
(ii) Sending StGQ to ∆Q gives an isomorphism of the lattice of subrepresentations of IndGB1
onto the lattice of upper sets in P(∆).
Proof. By the properties recalled before Theorem 3.2, (i) implies (ii). For (i) the proof is
given in [Her11, §10] when G is split, using results of Grosse-Kl¨onne [GK14]. The general
case is due to T. Ly [Ly15, beginning of §9].
We have variants of Proposition 3.5. If Q is a standard parabolic subgroup of G, the subrepresentations of IndGQ1 are the subrepresentations of IndGB1 contained in IndGQ1. So the
lattice of subrepresentations of IndGQ1 is isomorphic of the sublattice of upper sets in P(∆) consisting of subsets containing ∆Q; intersecting with ∆ \ ∆Q gives an isomorphism onto the lattice of upper sets in P(∆ \ ∆Q). More generally,
Proposition 3.6. Let P, Q be two standard parabolic subgroups of G with Q ⊂ P .
(i) The irreducible constituents of IndGPStPQ are the StGQ0 where Q0 ∩ P = Q, and each
occurs with multiplicity 1.
(ii) Sending StGQ0 to ∆Q0∩ (∆ \ ∆P) gives an isomorphism of the lattice of
subrepresenta-tions of IndGPStPQ onto the lattice of upper sets in P(∆ \ ∆P).
Proof. For (i), note that IndGPStPQis the quotient of IndGQ1 by the sum of its subrepresentations IndGQ01 for Q0 where Q ( Q0 ⊂ P and (i) is the content of [Ly15, Corollary 9.2]. The order
StGQ0 ≤ StGQ00 on the irreducible constituents corresponds (as it does in IndGB1) to ∆Q00 ⊂ ∆Q0.
Again (ii) follows for (i).
Remark 3.7. Note that P(∆ \ ∆P) does not depend on Q. The unique irreducible quotient of IndGP StPQ is StGQ, and its unique subrepresentation is StGQ0 where ∆Q0 = ∆Q∪ (∆ \ ∆P).
The next case where P1 = P, σ1= σ is a consequence of :
Proposition 3.8. Let P = M N be a standard parabolic subgroup of G and σ a supercuspidal
C-representation of M . Then the map X 7→ IndGP (σ)(e(σ) ⊗ X) gives an isomorphism of the
lattice of subrepresentations of IndP (σ)P 1 onto the lattice of subrepresentations of IndGPσ.
It has the immediate consequence:
Corollary 3.9. Sending IG(P, σ, Q) to ∆Q\ ∆P gives an isomorphism of the lattice of sub-representations of IndGP σ onto the lattice of upper sets in P(∆P (σ)\ ∆P).
The proposition 3.8 is proved in two steps, inducing first to P (σ) and then to G. In the first step we may as well assume that P (σ) = G:
Lemma 3.10. Let P = M N be a standard parabolic subgroup of G and σ a supercuspidal
C-representation of M such that P (σ) = G. Then the map X 7→ e(σ) ⊗ X gives an isomorphism of the lattice of subrepresentations of IndGP 1 onto the lattice of subrepresentations of e(σ) ⊗ IndGP1 ' IndGP σ.
Proof. By the classification theorem 3.1, the map X 7→ e(σ) ⊗ X gives a bijection between the
irreducible constituents of IndGP1 and those of e(σ) ⊗ IndGP 1. It is therefore enough to show that, for a parabolic subgroup Q of G containing P , the subrepresentation of e(σ) ⊗ IndGP 1 with cosocle e(σ) ⊗ StGQ is e(σ) ⊗ IndGQ1. Certainly, e(σ) ⊗ StGQ is a quotient of e(σ) ⊗ IndGQ1. Assume that e(σ) ⊗ StGQ is a quotient of e(σ) ⊗ IndGQ01 for some parabolic subgroup Q0 of G
containing P ; we want to conclude that Q0= Q. Recall from §2.2 that σ being supercuspidal, ∆P and ∆σ are orthogonal . Also, e(σ) is obtained by extending σ from M to G = M Mσ0 trivially on Mσ0. Upon restriction to Mσ0, therefore, e(σ) ⊗ IndGQ1 is a direct sum of copies of IndGQ1 whereas e(σ) ⊗ StGQ0 is a direct sum of copies of StGQ0. Thus there is a non-zero Mσ0
-equivariant map IndGQ1 → StGQ0. Let Misσ denote the isotropic part of the simply connected
covering of the derived group Mσ. Then Mσ0 is the image of Mσis in Mσ [AHHV17, II.4 Proposition]; moreover, as a representation of Mσis, IndGQ1 is simply IndMσis
Qis
σ 1 where Q
is
parabolic subgroup of Mσis corresponding to ∆Q∩ ∆σ, whereas StGQ0 is StM is σ Q0is σ . It follows that StMσis Q0is σ is a quotient of Ind Mis σ Qis σ 1, thus ∆Q ∩ ∆σ = ∆Q
0 ∩ ∆σ which implies ∆Q = ∆Q0 and
Q = Q0, since ∆Q and ∆Q0 both contain ∆P.
The second step in the proof of Proposition 3.8 is an immediate consequence of the following lemma, applied to P (σ) instead of P .
Lemma 3.11. Let P = M N be a standard parabolic subgroup of G. Let W be a finite length
smooth C-representation of M , and assume that for any irreducible subquotient Y of W ,
IndGPY is irreducible. The map Y 7→ IndGP Y from the lattice LW of subrepresentations of W to the lattice LIndG
PW
of subrepresentations of IndGP W is an isomorphism.
Proof. We recall from [Vig13, Theorem 5.3] that the functor IndGP has a right adjoint RGP and that the natural map Id → RGP IndGP is an isomorphism of functors. Let ϕ : LW → LIndG
PW
be the map Y 7→ IndGP Y and let ψ : LIndG
PW → LW be the map X 7→ R
G
PX. The composite ψ ◦ ϕ is a bijection. If ψ is injective, then ψ and ϕ are bijective, reciprocal to each other. To
show that ψ is injective, we show first that X ∈ LIndG
PW and R
G
PX ∈ LW have always the same length.
Step 1. An irreducible subquotient X of IndGPW has the form IndGPY for an irreducible
subquotient Y of W ; in particular, RGPX ' Y is irreducible. Thus, W and IndGPW have the
same length.
Step 2. Let X be a subquotient of IndGPW . Denote the length by lg(−). We prove that
lg(RGPX) ≤ lg(X), by induction on lg(X). If X 6= 0, insert X in an exact sequence 0 → X0 →
X → X00 → 0 with X00 irreducible; then the sequence 0 → RGPX0 → RG
PX → RGPX
00 is exact
and RGPX00 is irreducible. So lg(RGPX) ≤ lg(RGPX0) + 1 ≤ lg(X0) + 1 = lg(X). Step 3. Let X ∈ LIndG
PW. We deduce from the steps 1 and 2 that lg(R
G
PX) = lg(X). Indeed, the exact sequence 0 → X → IndGP W → (IndGP W )/X → 0 gives an exact sequence 0 → RGPX → W → RGP((IndGPW )/X). By Step 2, lg(RGPX) ≤ lg(X) and lg(RGP((IndGP W )/X)) ≤
lg((IndGP W )/X); by Step 1, lg(IndGPW ) = lg(W ), so we get equalities instead of inequalities.
We can show now that ψ is injective. Let X, X0 in LIndG
PW such that R
G
PX = RGPX0. Applying RGP to the exact sequence 0 → X ∩ X0 → X ⊕ X0 → X + X0 → 0 gives an exact
sequence 0 → RGP(X ∩ X0) → RGPX ⊕ RGPX0 → RG
P(X + X0) because RGP is compatible with direct sums. As RGP respects the length, the last map is surjective by length count. But then RGP(X + X0) = RGP(X) + RGP(X0) inside W . Hence RGP(X + X0) = RGPX = RGPX0. So
X = X0 = X + X0 by length preservation.
Remark 3.12. Note that lg(RGPX) = lg(X) for a subquotient X of IndGPW . Indeed, insert X in
an exact sequence 0 → X0 → X00 → X → 0 where X00 is a subrepresentation of IndG
PW . The exact sequence 0 → RGPX0 → RG PX 00 → RG PX and lg(RGPX 0) = lg(X0), lg(RG PX 00) = lg(X00)
give lg(RGPX) ≥ lg(X); with Step 2, this inequality is an equality.
We are now finally in a position to prove Theorem 3.2. It follows from Proposition 3.8 that X 7→ IndGP (σ)(e(σ) ⊗ X) gives an isomorphism of the lattice of subrepresentations of IndP (σ)P
1∩P (σ)St
M1∩P (σ)
Q (a quotient of the Ind P (σ)
P 1) onto the lattice of subrepresentations of IndGP (σ)(e(σ) ⊗ IndP (σ)P
1∩P (σ)St
M1∩P (σ)
Q ) isomorphic to Ind G
P1σ1. The desired result then follows
3.3. Twists by unramified characters. Recall the definition of unramified characters of
G. If XF∗(G) is the group of algebraic F -characters of G, we have a group homomorphism
HG : G → Hom(XF∗(G), Z) defined by HG(g)(χ) = valF(χ(g)) for g ∈ G and χ ∈ XF∗(G). The kernel 0G of HG is open and closed in G, and the image HG(G) has finite index in Hom(XF∗(G), Z). It is well known (see 2.12 in [HL17]) that0G is the subgroup of G generated
by its compact subgroups. A smooth character χ : G → C∗ is unramified if it is trivial on 0G; the unramified characters of G form the group of C-points of the algebraic variety
HomZ(HG(G), Gm).
Let σ1 be an irreducible admissible C-representation σ1 of M1 and we now examine the effect on IndGP
1σ1 of twisting σ1 by unramified characters of M1. As announced in §1.2,
we want to prove that for a general unramified character χ : M1 → C∗, the representation
IndGP1χσ1is irreducible. For that we translate the irreducibility criterion P (χ|Mσ) ⊂ P1 given
in Corollary 3.3 into more concrete terms. Note that χ|M is an unramified character of M . By Proposition 2.7, P (χ|Mσ) ⊂ P1 means that for each α ∈ ∆ \ ∆P1, χσ is non-trivial on
Z ∩ Mα0. Because χ|Mσ is supercuspidal, when α ∈ ∆ is not orthogonal to ∆P, χσ is not trivial on Z ∩ Mα0. Let ∆nr(σ) be the set of roots α ∈ ∆ \ ∆P1 orthogonal to ∆P, such that there exists an unramified character χα : M → C∗ such that χασ is trivial on Z ∩ Mα0; for α ∈ ∆nr(σ), choose such a χα.
Recall from [AHHV17, III.16 Proposition] that the quotient of Z ∩ Mα0 by its maximal compact subgroup is infinite cyclic; if we choose aα ∈ Z ∩ Mα0 generating the quotient, then χσ is trivial on Z ∩ Mα0 is and only if χ(aα) = χα(aα). We conclude:
Proposition 3.13. Let χ : M1 → C∗ be an unramified C-character of M1. Then IndGP1χσ1
is irreducible if and only if for all α ∈ ∆nr(σ) we have χ(aα) 6= χα(aα). The following corollary answers a question of J.-F. Dat.
Corollary 3.14. The set of unramified C-characters χ of M1 such that IndGP
1χσ1 is reducible
is a Zariski-closed proper subset of the space of unramified characters.
Indeed by the proposition, the reducibility set is the union, possibly empty, of hypersurfaces with equation χ(aα) = χα(aα) for α ∈ ∆nr(σ).
4. Admissibility
4.1. Generalities. Let H be a locally profinite group and let R be a commutative ring. When
R is noetherian, a subrepresentation of an admissible R-representation of H is admissible.
If H is locally pro-p and p is invertible in R, then taking fixed points under a pro-p open subgroup of H is an exact functor [Vig96, I.4.6], so for noetherian R a quotient of an admissible
R-representation of H is again admissible. This is not generally true, however when p = 0 in R, as the following example shows.
Example 4.1. Assume that p = 0 in R so that R is a Z/pZ-algebra. Let H be the additive
group (Z/pZ)N, with the product of the discrete topologies on the factors; it is a pro-p group.
The space C∞(H, R) (§2.2) can be interpreted as the space of functions H → R which depend only on finitely many terms of a sequence (un)n∈N ∈ H. The group H acts by translation yielding a smooth R-representation of H; if J is an open subgroup of H, the J -invariant functions in C∞(H, R) form the finitely generated free R-module of functions J \H → R. In particular, V = C∞(H, R) is an admissible R-representation of H. However the quotient of
form f ∈ HomZ/pZ(H, R) contained in V satisfies wf (v) − f (v) = f (w + v) − f (v) = f (w) for
v, w ∈ H so f produces an H-invariant vector in V /V0. Such linear forms make an infinite
rank free R-submodule of V and V /V0 cannot be admissible. That example will be boosted below in §4.2.
Lemma 4.2. Assume that R is noetherian. Let M be an module and t a nilpotent
R-endomorphism of M . Then M is finitely generated if and only if Ker t is.
Proof. If M is finitely generated so is its R-submodule Ker t, because R is noetherian.
Con-versely assume that Ker t is a finitely generated R-module; we prove that M is finitely gen-erated by induction over the smallest integer r ≥ 1 such that tr = 0. The case r = 1 is a tautology so we assume r ≥ 2. By induction, the R-submodule Ker tr−1 is finitely generated. As tr−1 induces an injective map M/ Ker tr−1 → Ker t of finitely generated image because R
is noetherian, the R-module M is finitely generated.
Lemma 4.3. Assume that R is noetherian. Let H be a locally pro-p group and J an open
pro-p subgroup of H. Let M be a smooth R-representation of H such that the multiplication pM by p on M is nilpotent. Then the following are equivalent:
(i) M is admissible;
(ii) MJ is finitely generated over R;
(iii) MJ∩ Ker pM is finitely generated over R/pR.
Proof. Clearly (i) implies (ii) and the equivalence of (ii) and (iii) comes from Lemma 4.2
applied to t = pM. Assume now (ii). To prove (i), it suffices to prove that for any open normal subgroup J0 of J , the R-module MJ0 is finitely generated. By Lemma 4.2, it suffices to do it for MJ0∩ Ker pM, that is, we can assume p = 0 in R. Now MJ
0
= HomJ0(R, M ) '
HomJ(R[J/J0], M ) as R-modules. The group algebra Fp[J/J0] has a decreasing filtration by two sided ideals Ai for 0 ≤ i ≤ r with A0 = Fp[J/J0], Ar = {0} and Ai/Ai+1 of dimension 1 over Fp with trivial action of J/J0. By tensoring with R we get an analogous filtration with Bi = R ⊗ Ai for R[J/J0]. By decreasing induction on i, we prove that HomJ(Bi, M ) is finitely generated over R. Indeed, the case i = r is a tautology, the exact sequence
0 → Bi+1→ Bi → Bi/Bi+1→ 0 gives an exact sequence
0 → HomJ(Bi/Bi+1, M ) → HomJ(Bi, M ) → HomJ(Bi+1, M )
and HomJ(Bi/Bi+1, M ) ' MJ is a finitely generated R-module by assumption. Since HomJ(Bi+1, M ) is finitely generated by induction, so is HomJ(Bi, M ) because R is
noe-therian. The case i = 0 gives what we want.
4.2. Examples. Let us now take up the case of a reductive connected group G = G(F ). Here the characteristic of F plays a role. When char(F ) = 0, G is an analytic p-adic group, in particular contains a uniform open pro-p subgroup, so that at least when R is a finite local Zp-algebra [Eme10] or a field of characteristic p [Hen09, 4.1 Theorem 1 and 2], a quotient of an admissible representation of G is still admissible. That does not survive when char(F ) = p, as the following example shows.
Example 4.4. An admissible representation of F∗ with a non-admissible quotient, when char(F ) = p > 0 and pR = 0.
The group 1 + PF is a quotient of F∗. Choose a uniformizer t of F . For simplicity assume that q = p. Then it is known that the map Q
(m,p)=1,m≥1Zp → 1 + PF sending (xm) to
Q
m(1+tm)xmis a topological group isomorphism. The group H of Example 4.1 is a topological quotient of F∗. When pR = 0 the admissible R-representation Cc∞(H, R) of H with the non-admissible quotient Cc∞(H, R)/Cc∞(H, R)H inflates to an admissible R-representation V of
F∗ containing the trivial representation V0= V1+PF with a non-admissible quotient V /V0.
That contrast also remains when we consider Jacquet functors. Let P = M N be a standard parabolic subgroup of G. Assume that R is noetherian. The parabolic induction IndGP :
M od∞R(M ) → M od∞R(G) respects admissibility [Vig13, Corollary 4.7]. Its left adjoint LGP respects admissibility when R is a field of characteristic different from p [Vig96, II.3.4]. More generally,
Proposition 4.5. Assume that R is noetherian and that p is invertible in V . Let V ∈ Mod∞R(G) such that for any open compact subgroup J of G, the R-module VJ has finite length. Then for any open compact subgroup JM of M , the R-module VJM
N has finite length. Proof. Assume that p is invertible in V . We recall first the assertions (i) and (ii) of the last
part of [Vig13]. Let (Kr)r≥0 be a decreasing sequence of open pro-p subgroups of G with an Iwahori decomposition with respect to P = M N , with Kr normal in K0, ∩Kr = {1}. We write κ : V → VN for the natural map and Mr = M ∩ Kr, Nr = N ∩ Kr, Wr = VKrN0. Let z ∈ Z(M ) strictly contracting N0 (subsection 2.5). Then we have
For any finitely generated submodule X of VMr
N there exists a ∈ N with zaX ⊂ κ(Wr). We prove now the proposition. As KrN0 is a compact open subgroup of G, the R-module Wr has finite length, say `. The R-modules κ(Wr) and zaX have finite length ≤ `, hence X also. This is valid for all X hence VMr
N has finite length ≤ `. We have zaV Mr
N ⊂ κ(Wr) ⊂ V Mr
N for some a ∈ N. The three R-modules have finite length hence κ(Wr) = VNMr. As any open compact subgroup JM of M contains Mr for r large enough, the proposition is proved. Remark 4.6. The proof is essentially due to Casselman [Cas], who gives it for complex
coef-ficients. The proof shows that VMr
N = κ(Wr) where Wr ⊂ VN0 for all r ≥ 0. This implies κ(VN0) = V
N because VN being smooth is equal to Sr≥0VNMr.
When R is artinian, any finitely generated R-module has finite length, so the proposition implies:
Corollary 4.7. LGP respects admissibility when R is artinian (in particular a field) and p is invertible in R.
Remark 4.8. This corollary was already noted by Dat [Dat09]. The corollary is expected to
be true for R noetherian when p is invertible in R. Using the theory of types, Dat proves it when G is a general linear group, a classical group with p odd, or a group of relative rank 1 over F .
Emerton has proved that LGP respects admissibility when R is a finite local Zp-algebra and char(F ) = 0 [Eme10]. But again, his proof does not survive when char(F ) = p > 0 and
pR = 0.
Example 4.9. An admissible representation of SL(2, F ) with a nonadmissible space of U
Assume char(F ) = p > 0 and pR = 0. Let B = T U the upper triangular subgroup of G = SL(2, F ) and identify T with F∗ via diag(a, a−1) 7→ a. Example 4.4 provides an admissible R-representation V of T containing the trivial representation V0 (the elements of V fixed by the maximal pro-p subgroup of T ), such that V /V0 is not admissible. The
representation IndGBV of G contains IndGBV0, which contains the trivial subrepresentation V00.
We claim that the quotient W = (IndGBV )/V00 is admissible and that WU is not admissible (as a representation of T ).
For the second assertion, it suffices to prove that WU = V /V0. The Steinberg representation St = IndGBV0/V00 of G is contained in W and W/St is isomorphic to IndGB(V /V0). We get an
exact sequence
StU → WU → (IndGB(V /V0))U → 0.
It is known that StU = 0 (see the more general result in Corollary 6.10 below). Hence the module WU is isomorphic to (IndGB(V /V0))U ' V /V0 [Vig13, Theorem 5.3].
We now prove the admissibility of W . Let U be the pro-p Iwahori subgroup of G, consist-ing of integral matrices in SL(2, OF) congruent modulo PF to the strictly upper triangular subgroup of SL(2, k). We prove that WU = StU, so W is admissible by Lemma 4.3, be-cause St is admissible. Let f ∈ IndGBV with a U -invariant image in W , hence for x ∈ U ,
there exists vx ∈ V0 with f (gx) − f (g) = vx for all g ∈ G. Put s =
0 1
−1 0
. Then
f (sx) − f (x) = f (sx) − vx − (f (x) − vx) = f (s) − f (1). Put v = f (s) − f (1) ∈ V . If x ∈ U , then sxs−1 ∈ U and f (sg) = f (sxs−1sg) = f (sxg). If x ∈ U ∩ U and z ∈ U we have f (sz) = f (z) + v = f (xz) + v = f (sxz). An easy matrix calculation shows that U is generated
by U ∩ U and U ∩ U , so the map z 7→ f (sz) from U to V is invariant under left multiplication by U . We have V0 = VU ∩T and U ∩ T is stable by conjugation by s. For t ∈ U ∩ T and z ∈ U we have f (sz) = f (stz) = sts−1f (sz) and f (z) = f (sz) − v = f (stz) − v = f (tz) = tf (z).
Therefore, f (sz) and f (z) lie in V0. But G is the union of BU and BsU , so f (g) ∈ V0 for all g ∈ G, which means f ∈ IndGBV0 and its image in W does belong to StU.
4.3. Admissibility and RGP. We turn to the main result of this section (theorem 1.3 of the introduction) for a general connected reductive group G and a standard parabolic subgroup
P = M N of G.
Lemma 4.10. Let V be a noetherian R-module, let t be an endomorphism of V , and view
V as a Z[T ]-module with T acting through t. Then the map f 7→ f (1) yields an isomorphism e from HomZ[T ](Z[T, T−1], M ) onto the submodule V∞ = ∩n≥0tnV of infinitely t-divisible elements.
Proof. A Z[T ]-morphism f : Z[T, T−1] → V is determined by the values mn = f (T−n) for
n ∈ N, which are only subject to the condition tmn+1 = mn for n ∈ N. Certainly f (1) = m0
is in V∞. Let us prove that e is surjective. As V is noetherian, there is some n ≥ 0 such that Ker tn+k = Ker tn for k ≥ 0. Let m ∈ V∞ and for k ≥ 0 choose mk such that m = tkmk. Then for k ≥ 0, mn+k− tmn+k+1 belongs to Ker tn+k so that tnmn+k= tn+1mn+k+1Putting µk = tnmn+k we have µk = tµk+1 and µ0 = m. Therefore e is surjective. By [Bou12, §2, No
2, Proposition 2], the action of t on V∞ being surjective is bijective because the R-module
V∞ is noetherian, so e is indeed bijective.
Theorem 4.11. Assume that R is noetherian and p is nilpotent in R. Then the functor
Proof. Let π be an admissible R-representation of G and we prove RGP(π) is admissible. By Lemma 4.3, we may replace π with Ker(p : π → π), hence we assume that p = 0 in R.
Recall that we have fixed a special parahoric subgroup K in §2.5. Take a finite extension F of Fp such that all absolute irreducible representations of K in characteristic p are defined over F. Then for any open pro-p subgroup J of K ∩ M , we have
RGP(π)J ⊂ RGP(F ⊗Fpπ)J = HomF[J ](F, RGP(F ⊗Fpπ))
= HomF[K∩M ](IndK∩MJ (F), RGP(F ⊗Fpπ)).
Since we have a filtration on IndK∩MJ (F) whose successive quotients are absolute irreducible representations, it is sufficient to prove that the R-module
HomF[K∩M ](V, RGP(F ⊗Fpπ)).
is finitely generated for any irreducible F-representation V of K ∩ M .
Put π1 = F ⊗Fp π. This is also admissible. Let V0 be an irreducible F-representation
of K which is P -regular [HV12, Definition 3.6] and (V0)N ∩K ' V . This V0 exists by the
classification of absolute irreducible representations of K ([HV12, Theorem 3.7], [AHHV17, III.10 Lemma]). Then by [HV12, Theorem 1.2] we have
IndGP(c-IndMK∩M(V )) ' HM(K ∩ M, V ) ⊗HG(K,V0)c-Ind
G
K(V0).
Hence
HomF[K∩M ](V, RPG(π1)) = HomF[M ](c-IndMK∩M(V ), RGP(π1))
= Hom F[G](Ind G P(c-IndMK∩M(V )), π1) = HomF[G](HM(K ∩ M, V ) ⊗H G(K,V0)c-Ind G K(V0), π1) = HomHG(K,V0)(HM(K ∩ M, V ), HomF[K](V0, π1)).
As HM(K ∩ M, V ) is a localization of HG(K, V0) at some τ ∈ ZG(K, V0), the R-module
HomH
G(K,V0)(HM(K ∩ M, V ), HomF[K](V0, π1))
identifies with
HomF[T ](F[T, T−1], HomF[K](V0, π1))
with T acting on HomF[K](V0, π1) through τ . Since the R-module HomF[K](V0, π1) is finitely
generated and R is noetherian, Lemma 4.10 show that HomF[T ](F[T, T−1], HomF[K](V0, π1))
is also a finitely generated R-module.
Remark 4.12. Using [OV17, Proposition 4.6] instead of [HV12, Corollary 1.3], the argument
works replacing K by a pro-p Iwahori subgroup. Note that the only irreducible representation of pro-p Iwahori subgroup in characteristic p is the trivial representation. So we may take F = Fp.
When R is noetherian, IndGP : Mod∞R(M ) → Mod∞R(G) respects admissibility and induces a functor IndG,aP : ModaR(M ) → ModaR(G) between the category of admissible representations. Emerton’s P -ordinary part functor OrdG
P is right adjoint to Ind G,a P . For V ∈ Mod ∞ R(G) admissible, (4) OrdGPV = (HomR[N ](Cc∞(N , R), V ))Z(M )−f,
is the space of Z(M )-finite vectors of HomR[N ](Cc∞(N , R), V ) with the natural action of M (the representation OrdG
P V of M is smooth) [Vig13, §8].
If RGP respects admissibility, the restriction of RGP to the category of admissible represen-tations is necessarily right adjoint to IndG,aP , hence is isomorphic to OrdG
P.
Corollary 4.13. Assume R noetherian and p nilpotent in R. Then RGP is isomorphic to the P -ordinary part functor OrdG
P on admissible R-representations of G.
Corollary 4.14. Assume that R is a field of characteristic p. Let V be an irreducible
admis-sible R-representation of G which is a quotient of IndGPW for some smooth R-representation W of M . Then V is a quotient of IndGPW0 for some irreducible admissible subquotient W0 of W .
The latter corollary was previously known only under the assumption that W admits a central character and R is algebraically closed [HV12, Proposition 7.10]. Its proof is as follows. By assumption, there is a non-zero M -equivariant map f : W → RGPV . By the
theorem RGPV is admissible so f (W ) contains an irreducible admissible subrepresentation W0
because char R = p [HV12, Lemma 7.9]. The inclusion of W0 into RG
PV gives a non-zero G-equivariant map IndGP W0 → V , so that V is a quotient of IndG
PW0.
Remark 4.15. When R is a field of characteristic 6= p and RGP respects admissibility, then Corollary 4.14 remains true.
Proof. It suffices to modify the proof of Corollary 4.14 as follows. We reduce to a finitely
generated R-representation W of M , by replacing W by the representation of M generated by the values of an element of IndGP W with non-zero image in V . An admissible quotient of W is also finitely generated, thus is of finite length [Vig96, II.5.10], and in particular, contains
an irreducible admissible subrepresentation W0. By the arguments in the proof of Corollary
4.14, V is a quotient of IndGPW0.
Let V ∈ Mod∞R(G). Obviously, OrdG
P(V ) given by the formula (4)depends only on the restriction of V to P , and LGPV = VN depends only on the restriction of V to P . We ask: Question 4.16. Does RPGV depend only on the restriction of V to P ?
To end this section we assume that R is noetherian and p is invertible in R and we compare
LGP and OrdGP. In the same situation than in Proposition 4.5, we take up the same notations. For V ∈ ModaR(G) we have the R-linear map
(5) ϕ 7→ κ(ϕ(1N0)) : Ord
G P(V )
eV
−→ LGP(V ) = VN,
where 1N0 is the characteristic function of N0. Replacing N0 by a compact open subgroup
JN ⊂ N multiplies eV by the generalized index [JN : N0] which is a power of p. Following
the action of m ∈ M which sends ϕ ∈ OrdGP(V ) to m ◦ ϕ ◦ m−1,
κ((mϕ)(1N0)) = κ(m(ϕ(1m−1N0m))) = [m
−1N
0m : N0]m(κ(ϕ(1N0))),
we get that eV is an R[M ]-linear map OrdGP(V ) → δ
−1
P LGP(V ), and that V 7→ eV defines on ModaR(G) a morphism of functors e : OrdGP → δP−1LGP. Here δP(m) = [mN0m−1 : N0] for m ∈ M .
Proposition 4.17. Assume R noetherian and p invertible in R. Let V ∈ Mod∞R(G) such
that for any open compact subgroup J of G, the R-module VJ has finite length. Then eV is an isomorphism.
Proof. 1) We recall the Hecke version of the Emerton’s functor [Vig13, §7, §8] for V ∈
ModaR(G). We fix an open compact subgroup N0 of N as in [Eme10, §3.1.1]. The monoid
M+⊂ M of m ∈ M contracting N0 acts on VN0 by the Hecke action:
(m, v) 7→ hm(v) = X n∈N0/mN0m−1
nmv : M+× VN0 → VN0.
We write IM
M+ : ModR(M+) → ModR(M ) for the induction, right adjoint of the restriction ResMM+ : ModR(M ) → ModR(M+). Let z ∈ Z(M ) strictly contracting N0 (subsection 2.5).
The map ΦV : OrdGP(V ) → (IMM+VN0)z −1−f
defined by
(6) ΦV(ϕ)(m) = (mϕ)(1N0)
is an isomorphism in ModaR(M ) (loc. cit. Proposition 7.5 restricted to the smooth and Z(M )-finite part, and Theorem 8.1 which says that the right hand side is admissible, hence is smooth and Z(M )-finite). For any r ≥ 0, Wr is stable by hz, the restriction from M to zZ gives a R[zZ]-isomorphism (7) ((IMM+VN0)z −1−f )Mr ' (IzZ zN(V N0Mr))z−1−f
(loc. cit. Remark 7.7 for z−1-finite elements, Proposition 8.2), the RHS of (7) is contained in IzzNZ(Wr), and we have the isomorphism
f 7→ (f (z−n))n∈N: IzzNZ(Wr) → {(xn)n≥0, xn∈ h∞z (Wr) = ∩n∈Nhnz(Wr), hz(xn+1) = xn} (loc. cit. Proposition 8.2, for the isomorphism Lemma 4.10).
2) The inclusion above is an equality (IzzNZ(VN0Mr))z−1−f = IzZ
zN(Wr), because the map
(8) f → f (1) : IzzNZ(Wr) → h
∞
z (Wr)
is an isomorphism: on the finitely generated R-module h∞z (Wr), hz is bijective as it is sur-jective (Lemma 4.10), hence any element f ∈ IzzNZ(Wr) is z−1-finite as (z−nf )(1) = f (z−n) for n ∈ N and a R-submodule of h∞z (Wr) is finitely generated.
Through the isomorphisms (6), (7), (8) the restriction of eV to (OrdP(V ))Mr translates into the restriction κr of κ to h∞z (Wr)
h∞z (Wr)−→ Vκr NMr.
3) The sequence Ker(hnz|Wr) is increasing hence stationary. Let n the smallest number such
that Ker(hnz|Wr) = Ker(hn+1z |Wr). By [Cas, III.5.3 Lemma, beginning of the proof of III.5.4 Lemma], Ker(κ|Wr) = Ker(h n z|Wr), h n z(Wr) ∩ Ker(hnz|Wr) = 0.
4) If the R-module Wrhas finite length, h∞z (Wr) = hnz(Wr) and Wr= hnz(Wr)⊕Ker(hnz|Wr).
Indeed, the sequence (hmz (Wr))m∈N is decreasing and lg(Wr) = lg(Ker(hmz |Wr)) + lg(h
m z (Wr)). Therefore κr is injective of image κ(Wr). As κ(Wr) = VNMr (proof of Proposition 4.5), κr is an isomorphism.
5) If the R-module Wr has finite length for any r ≥ 0, then κ(VN0) = VN (Remark 4.6)
